Lemma 8

(i) Every module admitting a finite presentation is finitely generated. (ii) If $A$ is a left Noetherian ring, every finitely generated left $A$-module admits a finite presentation. (iii) Every finitely generated projective module admits a finite presentation.

Assertion (i) follows trivially from the definitions. If $A$ is left Noetherian and there exists a surjective homomorphism $u : L_0 \to E$, where $L_0$ is a free left $A$-module with a finite basis, the kernel $R$ of $u$ is finitely generated (Algebra, Chapter VIII, , no. 1, Proposition 1 and no. 3, Proposition 7), hence there is a surjective homomorphism $u : L_1 \to R$, where $L_1$ is free and has a finite basis, and the exact sequence $L_1 \xrightarrow{v} L_0 \xrightarrow{u} E \to 0$ is a finite presentation of $E$; whence (ii)

Finally, suppose that $E$ is a finitely generated projective module; then it is a direct factor of a finitely generated free module $L_0$ (Algebra, Chapter II, §2, no. 2, Corollary to Proposition 4); the kernel $R$ of the surjective homomorphism $L_0 \to E$ is then isomorphic to a quotient of $L_0$ and hence finitely generated and the proof is completed as for (ii).